Page:1902 Encyclopædia Britannica - Volume 27 - CHI-ELD.pdf/796

(3) uniform tension at right angles to the radius vector of amount

$${\frac {1}{10}}gp(r^{2}/a)(1+3\rho )/(l-\rho ),$$

where g is the value of gravity at the surface. The corresponding strains consist of

(1) uniform contraction of all lines of the body of amount

$${\frac {1}{30}}K^{-1}gpa(3-\rho )/(1-\rho ),$$

(2) radial extension of amount ${\frac {1}{10}}k^{-1}gp(r^{2}/a)(l+\rho )/(1-\rho ),$

(3) extension in any direction at right angles to the radius vector of amount

$${\frac {1}{30}}k^{-1}gp+(r^{2}/a)/(1+\rho )/(1-\rho ),$$

where k is the modulus of compression. The volume is diminished by the fraction gpajbk of itself. The parts of the radii vectores within the sphere r = a{(3 - <r)/(3+3cr)}1/2 are contracted, and the parts without this sphere are extended. The application of the above results to the state of the interior of the earth is restricted by the circumstance that, unless the modulus of compression is much greater than that of any known material, the stresses and strains expressed above would, in a sphere of the size of the earth, greatly exceed the elastic limits. 25. In a spherical shell of homogeneous isotropic material, of internal radius rx and external radius r0, subjected to pressure pn on the outer surface, and px on the inner surface, the stress at any point distant r from the centre consists of (1) uniform tension in all directions of amount ^>1?r'13- r^0^-> o “ i /v) — 'fl 7*^ (2) radial pressure of amount — — 3 Tq r r* (3) tension in all directions at right angles to the radius vector of amount 3/°ri y» 3 , Vi-Pn r'0J1 3 _- rrx°3 ar3 # The corresponding strains consist of (1) uniform extension of all lines of the body of amount -iw 3& TV -r-i3 1 (2) radial contraction of amount ~ ^ (3) extension in all directions at right angles to the radius vector of amount _1_ P-Po r0W 4/a r03 - rx r3 ’ where p. is the modulus of rigidity of the material, =^E/(l + o-). The volume included between the7* ^two surfaces of the body is <2^ 3 7) increased by the fraction/ 1 3—2-^of itself, and the volume "(To ~ri) within the inner surface is increased by the fraction 3 3(pi-7?n) 3U 3 +1 pxr-??0r30 4/x r0 - rx &(r0 - rj ) of itself. For a shell subject only to internal pressure p the greatest extension is the extension at right angles to the radius at the inner surface, and its amount is _pp_ , J_ VV r03 - ri3 ^ 4/x rx ) ’ the greatest tension is zthe transverse tension at the inner surface, and its amount is p(r0 + rx)l(r(? - r^). 26. In the problem of a cylindrical shell under pressure a complication may arise from the effects of the ends ; but when the ends are free from stress the solution is very simple. With notation similar to that in § 25 it can be shown that the stress at a distance r from the axis consists of (1) uniform tension in all directions at right angles to the axis of amount Pr-P<?* r2_~2 > 2/,. 2 (2) radial pressure of amount Pl-W) VT; (3) hoop tension numerically equal to this radial pressure. The corresponding strains consist of (1) uniform extension of all lines of the material at right angles to the axis of amount l-o- pyr? -pnT<? E V-rx2 ’ (2) radial contraction of amount 1+q- P-i-Pn2 TyV E V-rx r2 ’ (3) extension along the circular filaments numerically equal to this radial contraction,

SYSTEMS (4) uniform contraction of the longitudinal filaments of amount 2 2°' Pr -Wq E 7y - rx2 For a shell subject only to internal pressure p the greatest extension is the circumferential extension at the inner surface, and its amount is P E W-rx2 J the greatest tension is the hoop tension at the inner surface, and its amount is p(r<? + r-?)l(r^ - ry ). 27. The results just obtained have been applied to gun construction; we may consider that one cylinder is heated so as to slip over another upon which it shrinks by cooling, so that the two form a single body in a condition of initial stress. We take P as the measure of the pressure between the two, and p for the pressure within the inner cylinder by which the system is afterwards strained, and denote by / the radius of the common surface. To obtain the stress at any point we superpose the Y2 ^ system consisting of radial pressure jp-b 3 and hoop tension o r- r0 -rx 2 r2 p'A o upon a system which, for the outer cylinder, consists

• -ri

a* 2 — ^.2 /pf<l 2 i m2 P-tx m and hoop tension P-5 -5 - r wof radial pressure r* ry ^'2 2 and, for the inner cylinder consists of radial pressure P-^ r 2 r2 + ry The hoop tension at the inner and hoop tension - P— surface is less than it would be for a tube of equal thickness without initial stress in the ratio 2/2 1-? 1. p rtf + rtf r’2 -1 k This shows how the strength of the tube is increased by the initial stress. 28. The problem of determining the distribution of stress and strain in a circular cylinder, rotating about its axis, has not yet been completely solved, but solutions have been obtained which are sufficiently exact for the two special cases of a thin disk and a long shaft. Suppose that a circular disk of radius a and thickness 21, and of density p, rotates about its axis with angular velocity w, and consider the following systems of superposed stresses at any point distant r from the axis and z from the middle plane: (1) uniform tension in all directions at right angles to the axis of amount |w2pa2(3 + cr), (2) radial pressure of amount |w2pr2(3+ <r), (3) pressure along the circular filaments of amount |w2pr2(l + 3<r), (4) uniform tension in all directions at right angles to the axis of amount ^co2p(Z2 - 322)<r(l + <r)/(l - cr). The corresponding strains may be expressed as (1) uniform extension of all filaments at right angles to the axis, of amount 1 ~E~ |wV2(3 + cr), (2) radial contraction of amount l-*3 o 2, —farpr (3) contraction along the circular filaments of amount l-ff2, 2 o —jr- $w2pr“, (4) extension of all filaments at right angles to the axis of amount ^ h a>2p(Z2 — 3^2)cr(l +cr), (5) contraction of the filaments normal to the plane of the disk of amount ^uPpa2 (3 + cr) - iW2pr2(l + cr) + u2p(l2 - 3s2)cr The greatest extension is the circumferential extension near the centre, and its amount is (3 + cr)(l -cr) 0 2 cr(l + cr) „ 2 8E u> + -g^ or pi . The longitudinal contraction is required to make the plane faces